Soft Computing

, Volume 23, Issue 2, pp 365–374 | Cite as

Homomorphism between ordered decision systems

  • Changzhong WangEmail author
  • Yang Huang
  • Xiaodong Fan
  • Mingwen Shao


Communication between information systems is an important topic in granular computing. The notion of homomorphism is viewed as a basic tool to study this kind of problems. This work studies basic properties of ordered decision systems under homomorphism. We first review consistent function related to ordered relation and introduce the notion of consistent function related to a universal subset. The relationship between the two kinds of consistent functions is given. Then, the relationships between ordered rough approximations and their images are discussed under consistent functions. Finally, ordered decision systems are divided into two classes: consistent and inconsistent ordered decision systems. For each type of decision system, some of its basic homomorphic properties are presented. It is proved that attribute reductions in an original system and its image system are equivalent to each other under the condition of homomorphism in each type of ordered decision system.


Consistent functions Relation mappings Ordered decision systems Homomorphism 



This work was supported by the National Natural Science Foundation of China under Grants 61572082, 61673396, and 61473111, the Foundation of Educational Committee of Liaoning Province (LZ2016003), the Natural Science Foundation of Liaoning Province (20170540012, 20170540004).

Compliance with ethical standards

Conflict of interest

Authors declare that they have no conflict of interest.

Human and animal rights disclosure

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Bryson N, Mobolurin A (1996) An action learning evaluation procedure for multiple criteria decision making problems. Eur J Oper Res 96:379–386CrossRefzbMATHGoogle Scholar
  2. Chen Y (2016) An adjustable multigranulation fuzzy rough set. Int J Mach Learn Cybern 7(2):267–274CrossRefGoogle Scholar
  3. Dai J, Xu Q (2013) Attribute selection based on information gain ratio in fuzzy rough set theory with application to tumor classification. Appl Soft Comput 13(1):211–221CrossRefGoogle Scholar
  4. Dai J, Hu H, Wu W, Qian Y, Huang D (2017) Maximal discernibility pairs based approach to attribute reduction in fuzzy rough sets. IEEE Trans Fuzzy Syst. Google Scholar
  5. Deng T, Chen Y, Xu W, Dai Q (2007) A novel approach to fuzzy rough sets based on a fuzzy covering. Inf Sci 177:2308–2326MathSciNetCrossRefzbMATHGoogle Scholar
  6. Dick S, Schenker A, Pedrycz W, Kandel A (2007) Regranulation: a granular algorithm enabling communication between granular worlds. Inf Sci 177:408–435MathSciNetCrossRefzbMATHGoogle Scholar
  7. Gong Z, Xiao Z (2010) Communicating between information systems based on including degrees. Int J Gen Syst 39(2):189–206MathSciNetCrossRefzbMATHGoogle Scholar
  8. Greco S, Matarazzo B, Slowinski R (2001) Rough sets theory for multicriteria decision analysis. Eur J Oper Res 129:11–47CrossRefzbMATHGoogle Scholar
  9. Greco S, Matarazzo B, Slowinski R (2002) Rough sets methodology for sorting problems in presence of multiple attributes and criteria. Eur J Oper Res 138:247–259MathSciNetCrossRefzbMATHGoogle Scholar
  10. Grzymala-Busse JW (1986) Algebraic properties of knowledge representation systems. In: Proceedings of the ACM SIGART international symposium on methodologies for intelligent systems, Knoxville, pp 432–440Google Scholar
  11. Grzymala-Busse JW, Sedelow WA Jr (1998) On rough sets, and information system homomorphism. Bull Pol Acad Sci Tech Sci 36(3&4):233–239zbMATHGoogle Scholar
  12. Iyer NS (2003) A family of dominance rules for multiattribute decision making under uncertainty. IEEE Trans Syst Man Cybern Part A 33:441–450CrossRefGoogle Scholar
  13. Kryszkiewski M (2001) Comparative study of alternative type of knowledge reduction in inconsistent systems. Int J Intell Syst 16:105–120CrossRefGoogle Scholar
  14. Lang G, Li Q, Guo L (2015a) Homomorphisms between covering approximation spaces. Fundam Inform 138(3):351–371MathSciNetzbMATHGoogle Scholar
  15. Lang G, Li Q, Guo L (2015b) Homomorphisms-based attribute reduction of dynamic fuzzy covering information systems. Int J Gen Syst 44(7–8):791–811MathSciNetCrossRefzbMATHGoogle Scholar
  16. Li D, Ma Y (2000) Invariant characters of information systems under some homomorphisms. Inf Sci 129:211–220MathSciNetCrossRefzbMATHGoogle Scholar
  17. Li Z, Liu Y, Li Q, Qin B (2016) Relationships between knowledge bases and related results. Knowl Inf Syst 49(1):171–19CrossRefGoogle Scholar
  18. Mi J, Zhang W (2004) An axiomatic characterization of a fuzzy generalization of rough sets. Inf Sci 160(1–4):235–249MathSciNetCrossRefzbMATHGoogle Scholar
  19. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356CrossRefzbMATHGoogle Scholar
  20. Pedrycz W, Vukovich G (2000) Granular worlds: representation and communication problems. Int J Intell Syst 15:1015–1026CrossRefzbMATHGoogle Scholar
  21. Pedrycz W, Bezdek JC, Hathaway RJ, Rogers GW (1998) Two nonparametric models for fusing heterogeneous fuzzy data. IEEE Trans Fuzzy Syst 6(3):411–425CrossRefGoogle Scholar
  22. Shao M, Zhang W (2005) Dominance relation and rules in an incomplete ordered information system. Int J Intell Syst 20:13–27CrossRefzbMATHGoogle Scholar
  23. She Y, He X (2014) Uncertainty measures in rough algebra with applications to rough logic. Int J Mach Learn Cybern 5(5):671–681CrossRefGoogle Scholar
  24. Slowinski R, Vanderpooten D (2000) A generalized definition of rough approximations based on similarity. IEEE Trans Knowl Data Eng 12(2):331–336CrossRefGoogle Scholar
  25. Sun L, Xu J, Tian Y (2012) Feature selection using rough entropy-based uncertainty measures in incomplete decision systems. Knowl Based Syst 36:206–216CrossRefGoogle Scholar
  26. Sun B, Ma W, Zhao H (2016) An approach to emergency decision-making based on decision-theoretic rough set over two universes. Soft Comput 20(9):3617–3628CrossRefGoogle Scholar
  27. Sun B, Ma W, Qian Y (2017a) Multigranulation fuzzy rough set over two universes and its application to decision making. Knowl Based Syst 123:61–74CrossRefGoogle Scholar
  28. Sun L, Zhang X, Xu J, Wang W, Liu R (2018) A gene selection approach based on the fisher linear discriminant and the neighborhood rough set. Bioengineered 9(1):144–151.
  29. Tsang ECC, Wang C, Chen D, Wu C, Hu Q (2013) Communication between information systems using fuzzy rough sets. IEEE Trans Fuzzy Syst 21(3):527–540CrossRefGoogle Scholar
  30. Wang C, Wu C, Wu Cong (2008) Communicating between information systems. Inf Sci 178:3228–3239MathSciNetCrossRefzbMATHGoogle Scholar
  31. Wang C, Chen D, Hu Q (2010) Some invariant properties of ordered information systems under homomorphism. Sci China Ser F Inf Sci 53:1816–1825MathSciNetCrossRefGoogle Scholar
  32. Wang C, Chen D, Sun B, Hu Q (2012) Communication between information systems with covering rough sets. Inf Sci 216:17–33MathSciNetCrossRefzbMATHGoogle Scholar
  33. Wang C, Shao M, He Q, Qian Y, Qi Y (2016) Feature subset selection based on fuzzy neighborhood rough sets. Knowl Based Syst 111(1):173–179CrossRefGoogle Scholar
  34. Wang C, He Q, Shao M, Hu Q (2017a) Feature selection based on maximal neighborhood discernibility. Int J Mach Learn Cybern. Google Scholar
  35. Wang C, Hu Q, Wang X, Chen D, Qian Y (2017b) Feature selection based on neighborhood discrimination index. IEEE Trans Neural Netw Learn Syst. Google Scholar
  36. Wu W (2008) Attribute reduction based on evidence theory in incomplete decision systems. Inf Sci 178:1355–1371MathSciNetCrossRefzbMATHGoogle Scholar
  37. Yang X, Qi Y, Yu D, Yu H, Yang J (2015) \(\alpha \)-Dominance relation and rough sets in interval-valued information systems. Inf Sci 294:334–347MathSciNetCrossRefzbMATHGoogle Scholar
  38. Yang Y, Chen D, Wang H, Tsang ECC, Zhang D (2017a) Fuzzy rough set based incremental attribute reduction from dynamic data with sample arriving. Fuzzy Sets Syst 312:66–86MathSciNetCrossRefzbMATHGoogle Scholar
  39. Yang Y, Chen D, Wang H (2017b) Active sample selection based incremental algorithm for attribute reduction with rough sets. IEEE Trans Fuzzy Syst 25(4):825–838CrossRefGoogle Scholar
  40. Yao YY (2008) Probabilistic rough set approximations. Int J Approx Reason 49(2):255–271CrossRefzbMATHGoogle Scholar
  41. Zhang H, Yang S (2016) Inclusion measure for typical hesitant fuzzy sets, the relative similarity measure and fuzzy entropy. Soft Comput 20(4):1277–1287MathSciNetCrossRefzbMATHGoogle Scholar
  42. Zhang H, Yang S (2017) Feature selection and approximate reasoning of large-scale set-valued decision tables based on alpha-dominance-based quantitative rough sets. Inf Sci 378:328–347CrossRefGoogle Scholar
  43. Zhao S, Chen H, Li C, Zhai M (2013) RFRR: robust fuzzy rough reduction. IEEE Trans Fuzzy Syst 21(5):825–841CrossRefGoogle Scholar
  44. Zhao S, Chen H, Li C, Du X, Sun H (2015) A novel approach to building a robust fuzzy rough classifier. IEEE Trans Fuzzy Syst 23(4):769–786CrossRefGoogle Scholar
  45. Zhu W (2007) Generalized rough sets based on relations. Inf Sci 177:4997–5011MathSciNetCrossRefzbMATHGoogle Scholar
  46. Zhu P, Wen Q (2010) Some improved results on communication between information systems. Inf Sci 180(18):3521–3531MathSciNetCrossRefzbMATHGoogle Scholar
  47. Zhu P, Wen Q (2011) Homomorphisms between fuzzy information systems revisited. Appl Math Lett 24(9):1548–1553MathSciNetCrossRefzbMATHGoogle Scholar
  48. Zhu P, Xie H, Wen Q (2014) A unified definition of consistent functions. Fundam Inform 135(3):331–340MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBohai universityJinzhouPeople’s Republic of China
  2. 2.Key Laboratory of Digital Publishing Big Data Mining Governance and Presentation Technology StandardBohai UniversityJinzhouPeople’s Republic of China
  3. 3.College of Computer and Communication EngineeringChina University of PetroleumQingdaoPeople’s Republic of China

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